Harder: Prove informally that the grammar G1 generates all
and only sentences in the set {an bn | n ≥ 1} .

Derive the string a a a b b b c c c using G2.

Harder: Prove informally that the grammar G2 generates all
and only sentences in the set {an bn cn | n ≥ 1} .

Given the grammar rules and lexicon shown below, derive the sentence This
is the house that Jack built. Draw the parse tree, labelling your subtrees
with the numbers of the grammar rules used to derive them.

Grammar rules:

and

Lexicon:

i. S → NP VP

v. NP → NP REL S

this: PRO

that: PRO, REL

ii. NP → PRON

vi. VP → V

is: V

Jack: NAME

iii. NP → ART N

vii. VP → V NP

the: ART

built: V

iv. NP → NAME

house: N

It was mentioned in lectures that context-sensitive grammar rules can be
defined either by requiring that they are of the form α →
β where the length of the string α is less than or equal to the
length of the string β (call these CS1 rules), or by requiring
that they are of the form λ A ρ → λ α ρ,
where λ and ρ are arbitrary, possibly null, strings of grammar
symbols (call these CS2 rules). The proof of this equivalence consists of
showing that rules of the form α → β can be converted into
sets of equivalent rules of the form λ A ρ → λ α ρ,
and the proof proceeds by induction on the length of the string β.

Begin this proof by showing that the CS1 rule A B → C D
may be replaced
by three CS2 rules. Hint: Introducea new non-terminal symbol
φ (i.e. a symbol not already in the alphabet of the grammar). Notice that
A B → A φ is a CS2 rule.

[For enthusiasts only]: Part (a), above, provides most of the
base-step of the inductive proof. Complete this proof by providing the
inductive step.